Method for determining structural deformation and internal forces of suspension bridge under lateral distributed live load

Abstract

The present invention discloses a calculation method for the internal force and deformation response of the entire bridge when the lateral distributed live load acts on the main girder of a suspension bridge. First, based on the known dead load state of the suspension bridge, the independent basic unknowns representing the key geometric forms and internal force parameters of the entire bridge are determined. Then, the responses of the main cable, main girder, bridge tower, and suspender are expressed in terms of the basic unknowns. Subsequently, according to the conditions of geometric compatibility, conservation of stress-free length, force balance, etc., control equations equal in number to the basic unknowns are listed. Finally, the unknowns are solved by the method of nonlinear programming, and the response of the entire bridge can be obtained. This method completely abandons the assumption in the traditional method that the suspenders are regarded as continuous membranes, and innovatively takes into account the effects of the following factors: longitudinal bending of the bridge tower, torsion of the bridge tower, elongation and inclination of the suspenders, longitudinal rigid body displacement of the main girder, etc., which is more in line with the actual stress state of the suspension bridge. The physical meaning of this method is clear, and the results can be obtained by adjusting several key parameters, which is convenient and fast.

Description

Technical Field

[0001] The present invention belongs to the field of bridge analysis theory, and particularly relates to a method for determining the structural deformation and internal force of a whole suspension bridge when a lateral distributed live load acts on the main girder of the suspension bridge. Background Art

[0002] The main load-bearing form of a suspension bridge is the tension of the main cable. Compared with other bridge types, it has a greater spanning ability and a greater flexibility, and is often prone to large deflection deformation. At present, there are many research results on the deflection response of suspension bridges under vertical loads, but few involve the research on lateral loads. There are only lateral wind-resistant bearings at the bridge towers in the lateral direction of the suspension bridge. Therefore, compared with the vertical stiffness, the lateral stiffness of the main girder is smaller, and the suspension bridge is more sensitive to lateral loads. Therefore, it is very necessary to study the deflection problem of long-span suspension bridges under lateral live loads.

[0003] The design theory of suspension bridges has experienced a development process from the initial elastic theory to the deflection theory, and finally to the current finite displacement theory. The initial elastic theory equation does not consider the beneficial effect of the deformation of the structure under load on the new equilibrium state, so the internal force response obtained is larger than the actual situation. Secondly, it does not utilize the beneficial effect of the dead load stiffness. Suspension bridges designed according to the elastic theory are often relatively bulky. Therefore, the elastic theory is not applicable to long-span suspension bridges. The solution of the deflection theory involves many nonlinear differential equations, and the calculation is relatively complex. Moreover, it cannot consider factors such as the spatial position change of the suspenders after bearing the load. With the maturity of computer technology, the finite element displacement theory method is generally used for the design and calculation of long-span bridges. The finite element method has a tendency to gradually replace the traditional deflection theory. However, the finite element method requires the establishment of a complex structural model, the operation process is relatively cumbersome, and the parameters of the model lack specific physical meanings, which is not conducive to engineers intuitively grasping the design information.

[0004] Therefore, a more rapid, more accurate and more in line with the actual stress state calculation method is needed for the deflection calculation of suspension bridges. Summary of the Invention

[0005] Object of the Invention: Aiming at the deficiencies of the above three calculation theories of suspension bridges, the present invention provides an analytical calculation method for the response of the entire suspension bridge when the main girder of the suspension bridge is subjected to a lateral distributed load. When a lateral distributed live load acts on the main girder, the main girder of the suspension bridge will generate a lateral deflection. The cable plane formed by the original main cables and suspenders will deform from a plane to a spatial cable plane, and the suspenders will no longer remain vertical; both the main girder and the main cables will generate lateral deflections; the bridge towers will generate lateral displacements and torsions in the longitudinal direction of the bridge; if the lateral live load is asymmetric, the main girder may undergo a longitudinal rigid body displacement; therefore, the displacements and internal forces of the entire bridge will become very complex. In the previous analytical calculation methods for the live load of suspension bridges, there are usually the following assumptions: (1) The suspenders are densely arranged and the suspender forces are membrane forces; (2) The suspenders do not tilt or elongate; (3) The main cables only undergo vertical deformations; (4) The bridge towers have no lateral displacements and torsional deformations; (5) The main girder does not undergo a longitudinal rigid body displacement. Although these assumptions are beneficial to simplify the calculation, they may deviate from the actual situation of the bridge and are no longer suitable for the trend of the increasing span of modern suspension bridges. Moreover, they cannot accurately solve the complex response of the entire suspension bridge when a lateral live load acts. In order to more accurately solve the response of the suspension bridge, this method fully considers factors such as the lateral displacement of the bridge tower, the torsion of the bridge tower, the longitudinal rigid body displacement of the main girder, the elongation of the suspenders, the change in the spatial position of the suspenders, and the non-uniform distribution of the lateral live load on the main girder. It is more in line with the actual use environment of the bridge and has stronger practicability.

[0006] Technical Solution: In order to achieve the above technical object, the present invention adopts the following technical solution:

[0007] A method for determining the structural deformation and internal forces of a suspension bridge under the action of a live load, comprising the following steps:

[0008] (1) Determine all the internal force and deformation characteristic parameters of the suspension bridge in the dead load state (initial state);

[0009] (2) Determine the independent basic unknowns representing the key geometric shapes and internal force parameters of the entire bridge;

[0010] (3) Express the response of the main cable with the basic unknowns, including the coordinates of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge;

[0011] (4) Express the response of the main girder with the basic unknowns, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder;

[0012] (5) Express the lateral displacement and torsional deformation of the bridge tower with the basic unknowns;

[0013] (6) Express the response of the suspenders with the basic unknowns, including the magnitudes of the forces of the suspenders in the three directions and the coordinates of the suspension points in the longitudinal, transverse, and vertical directions of the bridge;

[0014] (7) Establish control equations equal in number to the number of basic unknowns, including the conservation of the stress-free length of each catenary segment of the main cable; the conservation of the stress-free length of each hanger; the closure of the span and elevation difference of each span; the force balance of the main girder.

[0015] (8) Solve the control equation system to obtain the values of all basic unknowns.

[0016] (9) Substitute the basic unknowns into the above steps (2)(3)(4)(5)(6), and then obtain the responses of the entire bridge.

[0017] Furthermore, all the internal force and deformation characteristic parameters in step (1) mainly include the following three aspects:

[0018] (1) Establish geometric form coordinates; including the upper and lower suspension point coordinates of the hangers, the top coordinates of the bridge towers, the cable anchor point coordinates, the side span and main span lengths, the number and spacing of the hangers, and the rise-span ratios of the main cables in each span; (2) Set internal force parameters; including the internal forces of the main cable, main girder, and bridge towers, and the hanger forces; (3) Set physical characteristics; including the lateral bending and torsional flexibility coefficients of the bridge towers, the self-weight per unit length, elastic modulus, moment of inertia, cross-sectional area, Poisson's ratio, and density of each component.

[0019] Furthermore, in step (2), determine the basic unknowns representing the key geometric forms and internal force parameters of the entire bridge. Include the catenary equation parameters (dimensionless parameters related to the coordinate system selection) a L,u and a L,d of the left side span of the main cable, the catenary equation parameters a 1,u and a 1,d of the first catenary segment of the main span of the main cable, the catenary equation parameters a R,u and a R,d of the right side span of the main cable; the horizontal projected lengths L 1,u and L 1,d of the catenary segments of the left side span of the main cable, the horizontal projected lengths l 1,u ~l n+1,u and l 1,d ~l n+1,d of the first catenary segment of the main span of the main cable, the horizontal projected length L 3,u 、L 3,d of the catenary segment of the right side span of the main cable; the horizontal component forces H L,u and H L,d of the main cable in the left side span, the horizontal component forces H 1,u and H 1,d of the main cable in the first catenary segment of the main span, the horizontal component forces H R,u and H R,d of the main cable in the right side span; the transverse deflection angles α 1,u 、α 1,d of the main cable in the first catenary segment of the main span; the hanger forces P 1,u ~P n,u, P 1,d ~P n,d ; The change in the vertical support reaction of the main girder is ΔF L,z,u , ΔF L,z,d , ΔF R,z,u , ΔF R,z,d ; The horizontal support reaction of the wind-resistant support of the main girder is F L,y and F R,y ; The horizontal rigid body displacement ν of the main girder in the longitudinal direction of the bridge. Among them, the subscript "u" represents the upstream, "d" represents the downstream, and "n" represents the number of suspenders. There are a total of 4n + 27 independent basic unknowns.

[0020] Further, in step (3), the internal force and deformation response of the main cable are expressed in terms of basic unknowns, including the coordinates of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge. For the convenience of derivation, an overall coordinate system is established with the torsional center of the cross-section at the left end of the main girder in the initial state as the origin. The specific steps are as follows:

[0021] (3.1) The height difference between the two ends of the catenary of the main cable in the left side span is (here, the upstream and downstream are not distinguished):

[0022]

[0023] In the formula, c L = -H L / q, H L is the horizontal component force of the main cable in the left side span under the combined action of dead load and live load; q is the self-weight per unit length of the main cable, kN / m; a L is the parameter of the catenary equation in the left side span; L 1 is the horizontal projected length of the main cable in the left side span.

[0024] The stress-free length of the main cable in the left side span is:

[0025]

[0026] In the formula: E c and A c are the elastic modulus and cross-sectional area of the main cable respectively.

[0027] (3.2) The height difference between the two ends of the catenary of the main cable in the right side span is (here, the upstream and downstream are not distinguished):

[0028]

[0029] In the formula, c R = -H R / q, H R is the horizontal component force of the main cable in the right side span under the combined action of dead load and live load; a R The right side span is the parameter of the catenary equation; L 3is the horizontal projection length of the right side span across the main cable.

[0030] The stress-free length of the right side span main cable is:

[0031]

[0032] (3.3) The elevation difference between the two end points of the i-th catenary section of the main span main cable is:

[0033]

[0034] In the formula, α i is the angle between the plane where the i-th catenary section is located and the xoz plane; c i =-H i / q, H i is the horizontal component force of the i-th main cable under the combined action of dead load and live load; a i is the parameter of the i-th catenary equation of the main span; l i is the horizontal projection length of the i-th main cable of the main span.

[0035] The stress-free length of the main cable of the i-th catenary section can be expressed as:

[0036]

[0037] (3.4) Under the combined action of dead load and live load, the recurrence relations of the catenary parameters a, the horizontal force H of the main cable, and the angle α between the plane where the catenary is located and the xoz plane between adjacent catenary sections of the main span are expressed as:

[0038]

[0039]

[0040]

[0041] In the formula, P i,x , P i,y , P i,z are the component forces of the hanger force P i in the x, y, and z directions respectively.

[0042] Therefore, given a, H, and α of the first catenary section of the main span, the catenary equation parameters of the remaining sections can be expressed.

[0043] (3.5) The coordinates of the suspension points on the main cable:

[0044]

[0045] Y c,i =±b h -△y c,i

[0046]

[0047] Wherein, X c,i , Y c,i and Z c,i are respectively the X, Y, and Z coordinates of the i-th upper suspension point in the global coordinate system; Δx B is the offset of the bridge tower; b h is the transverse distance from the suspender to the longitudinal center line of the main girder; H B is the elevation of the left bridge tower.

[0048] Furthermore, the specific steps for expressing the internal force and deformation response of the main girder with basic unknowns in step (4) are as follows:

[0049] (4.1) Vertical deflection of the main girder:

[0050] The main girder is divided into n + 1 segments along the longitudinal direction of the bridge by suspenders, and the main girder receives a total of n + 2 forces vertically. These include the increments of the vertical reaction forces at the left and right ends of the main girder, ΔF L,z , ΔF R,z and n pairs of increments of vertical suspender forces. At the m-th suspender, the vertical deflection of the main girder can be expressed as:

[0051]

[0052] Wherein, x m represents the horizontal distance from the m-th pair of suspenders to the left end of the main girder; x i represents the horizontal distance from the i-th pair of suspenders to the left end of the main girder; ΔP i,z represents the increment of the vertical suspender force of the i-th pair of suspenders; E b is the elastic modulus of the main girder; I z is the vertical flexural moment of inertia of the main girder; C m,1 and C m,2 represent the constant terms after integrating the vertical bending moment of the main girder.

[0053] The recurrence relationships between the indefinite integral constants C m,1 and C m+1,1 of the m-th segment and the (m + 1)-th segment of the main girder, as well as between C m,2 and C m+1,2 are as follows:

[0054]

[0055]

[0056] Wherein, ΔP m,z represents the increment of the vertical suspender force of the m-th pair of suspenders

[0057] The indefinite integral constants of the first segment of the main girder are as follows:

[0058]

[0059] C 1,2 = 0

[0060] (4.2) Lateral deflection of the main girder:

[0061] Along the longitudinal direction of the bridge, the main girder is divided into n + 1 segments by the suspenders, and the main girder receives a total of n + 2 forces in the transverse direction of the bridge. These include the lateral support reactions F L,y 、F R,y at the left and right ends of the main girder and n pairs of increments of the lateral suspender forces. At the m-th suspender, the lateral deflection of the main girder can be expressed as:

[0062]

[0063] In the formula, ΔP m,y represents the increment of the lateral suspender force of the m-th pair of suspenders; I y is the lateral flexural moment of inertia of the main girder; D m,1 and D m,2 represent the constant terms after the integration of the lateral bending moment of the main girder; κ(x) is a function generated by the double integration of the bending moment.

[0064] The recurrence relations between the indefinite integral constants D m,1 and D m+1,1 of the m-th segment and the (m + 1)-th segment of the main girder, as well as between D m,2 and D m+1,2 are as follows:

[0065]

[0066]

[0067] The indefinite integral constants of the first segment of the main girder are as follows:

[0068]

[0069] D 1,2 = 0

[0070] (4.3) Torsional deformation of the main girder:

[0071] Due to the lateral deflection of the main girder, the vertical components of the suspender forces on the upstream and downstream sides are no longer symmetric, and the main girder will twist under the action of the asymmetric vertical suspender forces. The torque within the m-th segment of the main girder is:

[0072]

[0073] In the formula, b s and b hThey are the transverse distances from the vertical supports at the beam ends and each hanger rod to the longitudinal center line of the main beam; ΔP i,z,u and ΔP i,z,d are the hanger rod forces at the upstream and downstream sides of the i-th hanger rod of the main beam in the longitudinal direction of the bridge, respectively.

[0074] The rotation angle of the right end of the m-th main beam segment relative to the left end of this beam segment is:

[0075]

[0076] In the formula, d m is the length of the m-th main beam segment; G is the shear modulus of the main beam; I P is the polar moment of inertia of the main beam.

[0077] The vertical deflections caused by torsion at the i-th hanger points on the upstream and downstream sides of the main beam are respectively and

[0078] Furthermore, the specific steps for expressing the internal forces and deformation responses of the bridge tower with basic unknowns in step (5) are as follows:

[0079] The offset Δ B of the center of the left bridge tower top towards the mid-span after deformation is expressed as:

[0080] △ B =(H 1,u +H 1,d -H L,u -H L,d )·δ B

[0081] In the formula, δ B is the lateral bending flexibility coefficient of the left tower, m / kN; H 1,u and H 1,d are the horizontal component forces of the first main cable segments on the upstream and downstream sides of the main span respectively; H L,u and H L,d are the horizontal component forces of the main cables on the upstream and downstream sides of the left side span respectively.

[0082] The torsional angle (positive in the counterclockwise direction) θ B of the left bridge tower after deformation is expressed as:

[0083] θ B =(H 1,d -H L,d -H 1,u +H L,u )b t ·ζ B

[0084] In the formula, ζ B is the torsional flexibility coefficient of the left tower; b tThe horizontal distance of each limb tower column from the central axis.

[0085] The longitudinal offsets at the tops of the upstream and downstream tower columns of the left tower are respectively:

[0086] △ B,u =△ B -θ B ·b t

[0087] △ B,d =△ B +θ B ·b t

[0088] The offset Δ of the center of the top of the right bridge tower towards the mid-span after deformation C is expressed as:

[0089] △ C =(H n,u +H n,d -H R,u -H R,d )·δ C

[0090] where δ C is the lateral bending flexibility coefficient of the right tower, m / kN; H n,u and H n,d are respectively the horizontal component forces of the nth section of the main cable in the upstream and downstream of the main span; H R,u and H R,d are respectively the horizontal component forces of the main cable in the upstream and downstream of the right side span.

[0091] The torsional angle (positive in the counterclockwise direction) θ of the right bridge tower after deformation C is expressed as:

[0092] θ C =(H R,d -H n,d -H R,u +H n,u )b t ·ζ C

[0093] where ζ C is the torsional flexibility coefficient of the right tower.

[0094] The longitudinal offsets at the tops of the upstream and downstream tower columns of the right tower are respectively:

[0095] △ C,u =-△ C -θ C ·b t

[0096] △ C,d =-△ C+θ C ·b t

[0097] Furthermore, in step (6), the magnitudes of the three-direction components of the hanger force and the spatial coordinates of the lower suspension point are expressed in terms of the basic unknowns, including the longitudinal, transverse, and vertical coordinates in the bridge direction:

[0098] Under the combined action of the dead load and the live load, the coordinates of the lower suspension point of any hanger in the global coordinate system can be expressed as:

[0099] X b,i = x i + v

[0100] Y b,i = ±b h - Δy b,i

[0101]

[0102] In the formula, X b,i , Y b,i and Z b,i are the X, Y, and Z coordinates of the i-th lower suspension point in the global coordinate system respectively; v is the longitudinal rigid body displacement of the main girder; Δy b,i is the transverse deflection of the main girder at the i-th pair of hangers; Δz b,i is the vertical deflection of the main girder at the i-th pair of hangers; the ± here is applicable to the upstream and downstream respectively; H B is the elevation of the left bridge tower.

[0103] The transverse inclination angle θ i and the vertical inclination angle of the hanger can be expressed respectively as follows:

[0104]

[0105]

[0106] The components of the hanger force in the three directions can be expressed respectively as:

[0107]

[0108]

[0109]

[0110] Therefore, the magnitudes of the three-direction components of the hanger and the spatial coordinates can be expressed in terms of the basic unknowns.

[0111] Furthermore, the specific steps for establishing the control equations equal to the number of basic unknowns in step (7) are as follows:

[0112] (7.1) The stress-free length of each catenary segment of the main cable is conserved:

[0113] S c,L = S' c,L

[0114] S c,i = S' c,i , 1 ≤ i ≤ n + 1

[0115] S c,R = S' c,R

[0116] Wherein, S' c,L and S' c,R are the stress-free lengths of the main cables of the left and right side spans in the initial state respectively; S' c,i is the stress-free length of the i-th catenary segment of the main cable of the main span in the initial state.

[0117] (7.2) The stress-free length of each hanger is conserved:

[0118]

[0119] S h,i = S' h,i

[0120] Wherein, E h and A h are the elastic modulus and cross-sectional area of the hanger respectively; S' h,i is the stress-free length of the i-th hanger in the initial state; S h,i is the stress-free length of the i-th hanger after the combined action of the dead load and the live load.

[0121] (7.3) Taking the upstream side as an example, the span and elevation difference of each span are closed. The control equation for the downstream side can be obtained by changing u to d in the following formula:

[0122] Closure of the span of each span:

[0123] L' 1,u = L 1,u - △ B,u

[0124]

[0125] L' 3,u = L 3,u - △ C,u

[0126] Wherein, L' 1,u 、L' 2,u and L' 3,u are the horizontal projection lengths of the catenaries of the left side span, the main span and the right side span respectively after being subjected to the lateral load.

[0127] The elevation closure of each span (taking the upstream side as an example):

[0128] △z L,u = H B - H A

[0129]

[0130] △z R,u = H C - H D

[0131] The cross - bridge horizontal distance closure at both ends of the main cable of the main span (taking the upstream side as an example):

[0132]

[0133] (7.4) Force balance of the main girder:

[0134] After the main girder is under the combined action of dead load and live load, the force balance in the x, y, and z directions:

[0135]

[0136]

[0137]

[0138] After the main girder is under the combined action of dead load and live load, the moment balance about the x, y, and z axes:

[0139]

[0140]

[0141]

[0142] The torsional angle of the right end of the main girder about the x - axis is 0:

[0143]

[0144] Furthermore, the specific steps for solving the control equations at one time in step (8) to obtain the values of the basic unknowns are as follows:

[0145] A total of 4n + 27 control equations are obtained in step (7). Move the right - hand side of each equation to the left - hand side to get a form like f i = 0 (i = 1, 2... n), and then rewrite it as a sum of squares to get a form like f i 2in the form of =0 (i = 1, 2... n), and finally add them up to obtain the objective function:

[0146]

[0147] Use the method of programming solution to solve the programming of the objective function, and solve the values of the 4n + 27 basic unknowns in step (7).

[0148] Furthermore, in step (9), substitute the basic unknowns into steps (3), (4), (5), and (6), and then obtain the specific steps of the full-bridge response as follows:

[0149] Substitute the values of the basic unknowns obtained in step (8) back into step (3) to determine the response of the main cable under live load, including the displacements of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge;

[0150] Substitute the values of the basic unknowns obtained in step (8) back into step (4) to determine the response of the main girder under live load, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder;

[0151] Substitute the values of the basic unknowns obtained in step (8) back into step (5) to determine the deformation and internal force of the suspenders under live load;

[0152] Substitute the values of the basic unknowns obtained in step (8) back into step (6) to determine the position and spatial position of the suspenders under live load, including the offsets of the suspenders in the longitudinal, transverse, and vertical directions.

[0153] Beneficial effects:

[0154] Adopting the above technical solutions, the present invention has the following beneficial effects: The present invention abandons many assumptions in the previous calculation theories, fully considers factors such as the side displacement of the bridge tower, the torsion of the bridge tower, the longitudinal rigid body displacement of the main girder, the elongation of the suspenders, the change in the spatial position of the suspenders, and the uneven distribution of the lateral live load on the main girder, etc., which is more in line with the actual situation of the suspension bridge and has stronger practicability; The method of the present invention has a clear physical meaning, and the results can be quickly obtained by adjusting several key parameters, which is convenient and fast. Description of the drawings

[0155] Figure 1 It is a schematic diagram of the suspension bridge under the lateral load in a specific embodiment;

[0156] Figure 2 It is a schematic diagram of the main cable under the dead load state in a specific embodiment;

[0157] Figure 3 It is a schematic diagram of the side-span main cable under the combined action of the dead load and the live load in a specific embodiment;

[0158] Figure 4 Schematic diagram of the main span main cable under the combined action of dead load and live load in a specific embodiment;

[0159] Figure 5 Schematic diagram of the i-th catenary in a specific embodiment;

[0160] Figure 6 Schematic diagram of the force analysis of the common node of two adjacent catenaries in a specific embodiment;

[0161] Figure 7 Schematic diagram for calculating the vertical deflection of the main girder under the combined action of dead load and live load in a specific embodiment;

[0162] Figure 8 Schematic diagram for calculating the lateral deflection of the main girder under the combined action of dead load and live load in a specific embodiment;

[0163] Figure 9 Schematic diagram for calculating the torsional deformation of the main girder under the combined action of dead load and live load in a specific embodiment;

[0164] Figure 10 Schematic diagram for calculating the deformation of the bridge tower under the combined action of dead load and live load in a specific embodiment;

[0165] Figure 11 Lateral inclination angle θ in a specific embodiment i and vertical inclination angle Schematic diagram; Detailed implementation mode

[0166] The following further clarifies the application process of the present invention in combination with specific embodiments. These embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. After reading the present invention, various equivalent modifications made by those skilled in the art to the present invention fall within the scope defined by the appended claims of this application.

[0167] A method for determining the structural deformation and internal force of a suspension bridge under live load according to the present invention, the method comprising the following steps:

[0168] (1) Determine all internal force and deformation characteristic parameters of the suspension bridge in the dead load state (initial state);

[0169] (2) Determine the independent basic unknowns representing the key geometric shapes and internal force parameters of the whole bridge;

[0170] (3) Express the main cable response with the basic unknowns, including the displacements of the main cable suspension points in the longitudinal, vertical and transverse directions of the bridge;

[0171] (4) Express the response of the main girder with the basic unknowns, including the longitudinal drift, transverse deflection, vertical deflection and torsional deformation of the main girder;

[0172] (5) Express the lateral displacement and torsional deformation of the bridge tower in terms of the basic unknowns;

[0173] (6) Express the responses of the suspenders in terms of the basic unknowns, including the magnitudes of the forces in three directions of the suspenders and the coordinates of the suspension points in the longitudinal, transverse, and vertical directions of the bridge;

[0174] (7) Establish control equations equal in number to the basic unknowns, including the conservation of the stress-free lengths of the catenary segments of the main cable; the conservation of the stress-free lengths of each suspender; the closure of the span lengths and height differences of each span; the force balance of the main girder;

[0175] (8) Solve the control equation system to obtain the values of all the basic unknowns;

[0176] (9) Substitute the basic unknowns into steps (2), (3), (4), (5), and (6) above, and then obtain the responses of the entire bridge;

[0177] The specific implementation method is as follows:

[0178] The first step: Identify all the internal force and deformation characteristic parameters of the suspension bridge in the dead load state (initial state). In the initial state, the main cable of the suspension bridge is as Figure 2 shown, and the main internal force and deformation characteristic parameters mainly include the following three aspects:

[0179] (1) Geometric shape. It includes the upper and lower suspension point coordinates of the suspenders; the top coordinates of the bridge tower; the anchor point coordinates of the main cable; the side span and main span lengths; the number and spacing of the suspenders, the rise-span ratio of the main cable of each span, etc. (2) Internal force parameters. It includes the internal forces of the main cable, main girder, and bridge tower; the suspender forces, etc. (3) Physical characteristics. It includes the lateral bending and torsional flexibility coefficients of the bridge tower; the self-weight per unit length, elastic modulus, moment of inertia, cross-sectional area, Poisson's ratio, density, etc. of each component.

[0180] The second step: Determine the independent basic unknowns representing the key geometric shapes and internal force parameters of the entire bridge.

[0181] It includes the catenary equation parameters (dimensionless parameters related to the coordinate system selection) a L,u and a L,d of the left side span of the main cable, the catenary equation parameters a 1,u and a 1,d of the first catenary segment of the main span of the main cable, the catenary equation parameters a R,u and a R,d of the right side span of the main cable; the horizontal projected lengths L 1,u and L 1,d of the catenary segments of the left side span of the main cable, the horizontal projected lengths l 1,u ~l n+1,u and l 1,d ~l n+1,d, the horizontal projection length L of the catenary segment of the main cable in the right side span 3,u , L 3,d ; the horizontal component force H of the main cable in the left side span L,u and H L,d , the horizontal component force H of the main cable of the first catenary segment in the main span 1,u and H 1,d , the horizontal component force H of the main cable in the right side span R,u and H R,d ; the lateral deflection angle α of the main cable of the first catenary segment in the main span 1,u , α 1,d ; the hanger force P 1,u ~ P n,u , P 1,d ~ P n,d ; the change amount ΔF of the vertical support reaction of the main girder L,z,u , ΔF L,z,d , ΔF R,z,u , ΔF R,z,d ; the horizontal support reaction F of the wind-resistant support of the main girder L,y and F R,y ; the horizontal rigid body displacement ν of the main girder along the bridge axis. Among them, the subscript "u" represents the upstream, "d" represents the downstream, and "n" represents the number of hangers. There are a total of 4n + 27 independent basic unknowns.

[0182] Step 3: Express the main cable response with the basic unknowns, including the coordinates of the main cable suspension points in the longitudinal, vertical, and transverse directions of the bridge axis.

[0183] (1) Under the combined action of dead load and live load, the side span main cable is as Figure 3 shown. For the convenience of derivation, an overall coordinate system is established with the torsional center of the cross-section at the left end of the main girder in the initial state as the origin. The elevation difference between the two ends of the catenary of the main cable in the left side span is (here, the upstream and downstream are not distinguished):

[0184]

[0185] In the formula, c L = -H L / q, H L is the horizontal component force of the main cable in the left side span under the combined action of dead load and live load; q is the self-weight per unit length of the main cable, kN / m; a L is the parameter of the catenary equation in the left side span; L 1 is the horizontal projection length of the main cable in the left side span.

[0186] The stress-free length of the main cable in the left side span is:

[0187]

[0188] In the formula: E c and A care the elastic modulus and cross-sectional area of the main cable respectively.

[0189] (2) The elevation difference between the two end points of the catenary of the main cable in the right side span (without distinguishing upstream and downstream here) is:

[0190]

[0191] In the formula, c R =-H R / q, H R is the horizontal component force of the main cable in the right side span under the combined action of dead load and live load; a R is the parameter of the catenary equation for the right side span; L 3 is the horizontal projected length of the main cable in the right side span.

[0192] The stress-free length of the main cable in the right side span is:

[0193]

[0194] (3) The main cable of the main span under the combined action of dead load and live load is as Figure 4 shown. The elevation difference between the two end points of the i-th catenary segment of the main cable of the main span (as Figure 5 shown) is:

[0195]

[0196] In the formula, α i is the angle between the plane where the i-th catenary segment is located and the xoz plane; c i =-H i / q, H i is the horizontal component force of the i-th main cable segment under the combined action of dead load and live load; a i is the parameter of the i-th catenary equation of the main span; l i is the horizontal projected length of the i-th main cable segment of the main span.

[0197] The i-th catenary segment is as Figure 5 shown. Its stress-free length of the main cable can be expressed as:

[0198]

[0199] (4) Under the combined action of dead load and live load, by analyzing the forces at the common nodes of two adjacent catenary segments (as Figure 6 shown), the recurrence relationships of the catenary parameter a, the horizontal force H of the main cable, and the angle α between the plane where the catenary is located and the xoz plane between adjacent catenary segments of the main span are expressed as:

[0200]

[0201]

[0202]

[0203] In the formula, P i,x , P i,y , P i,z are the component forces of the suspension rod force P i in the x, y, and z directions respectively.

[0204] Therefore, given the a, H, and α of the first catenary segment of the main span, the catenary equation parameters of the remaining segments can be expressed.

[0205] (5) Coordinates of the hanging points on the main cable:

[0206]

[0207] Y c,i = ±b h - △y c,i

[0208]

[0209] In the formula, X c,i , Y c,i and Z c,i are the X, Y, and Z coordinates of the i-th hanging point in the global coordinate system respectively; Δx B is the offset of the bridge tower; b h is the transverse distance from the suspension rod to the longitudinal center line of the main beam; H B is the elevation of the left bridge tower.

[0210] Fourth step: Express the response of the main beam with the basic unknowns, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main beam.

[0211] (1) Vertical deflection of the main beam:

[0212] The main beam is divided into n + 1 segments along the longitudinal direction by the suspension rods. The main beam receives a total of n + 2 forces in the vertical direction, as Figure 7 shown. It includes the increments of the left and right vertical support reactions of the main beam, ΔF L,z , ΔF R,z and n pairs of increments of the vertical suspension rod forces. At the m-th suspension rod, the vertical deflection of the main beam can be expressed as:

[0213]

[0214] In the formula, x m represents the horizontal distance from the m-th pair of suspension rods to the left end of the main beam; x i represents the horizontal distance from the i-th pair of suspension rods to the left end of the main beam; ΔP i,z represents the increment of the vertical suspension rod force of the i-th pair of suspension rods; E bis the elastic modulus of the main girder; I z is the vertical flexural moment of inertia of the main girder; C m,1 and C m,2 represent the constant term after integrating the vertical bending moment of the main girder.

[0215] The indefinite integral constants C m,1 of the m-th main girder segment and the (m + 1)-th main girder segment, and C m+1,1 as well as C m,2 and C m+1,2 have the following recurrence relations:

[0216]

[0217]

[0218] The indefinite integral constants of the first main girder segment are as follows:

[0219]

[0220] C 1,2 = 0

[0221] (2) Lateral deflection of the main girder:

[0222] Along the longitudinal direction of the bridge, the main girder is divided into n + 1 segments by suspenders. The main girder receives a total of n + 2 forces in the transverse direction of the bridge, as Figure 8 shown. It includes the lateral support reactions F L,y , F R,y at both ends and n pairs of lateral suspender force increments. At the m-th suspender, the lateral deflection of the main girder can be expressed as:

[0223]

[0224] In the formula, ΔP m,y represents the lateral suspender force increment of the m-th pair of suspenders; I y is the lateral flexural moment of inertia of the main girder; D m,1 and D m,2 represent the constant terms after integrating the lateral bending moment of the main girder; κ(x) is the function generated by integrating the bending moment twice.

[0225] The indefinite integral constants D m,1 of the m-th main girder segment and the (m + 1)-th main girder segment, and D m+1,1 as well as D m,2 and D m+1,2 have the following recurrence relations:

[0226]

[0227]

[0228] The indefinite integral constants of the first main girder segment are as follows:

[0229]

[0230] D 1,2 = 0

[0231] (3) Twisting deformation of the main girder:

[0232] Due to the lateral deflection of the main girder, the vertical components of the hanger forces on the upstream and downstream sides are no longer symmetric. Under the action of the asymmetric vertical hanger forces, the main girder will twist, as Figure 9 shown. The torque in the m-th segment of the main girder is:

[0233]

[0234] In the formula, b s and b h are the transverse distances from the vertical support at the beam end and each hanger to the longitudinal center line of the main girder respectively; ΔP i,z,u and ΔP i,z,d are the hanger forces on the upstream and downstream sides at the i-th hanger of the main girder in the longitudinal direction of the bridge respectively.

[0235] The rotation angle of the right end of the m-th segment of the main girder relative to the left end of this beam segment is:

[0236]

[0237] In the formula, d m is the length of the m-th segment of the main girder; G is the shear modulus of the main girder; I P is the polar moment of inertia of the main girder.

[0238] The vertical deflections generated by torsion at the i-th hanger on the upstream and downstream sides of the main girder are respectively and

[0239] Step 5: Express the lateral displacement and twisting deformation of the bridge tower with the basic unknowns.

[0240] The deformation of the bridge tower under the combined action of the dead load and the live load is as Figure 10 shown. The offset Δ B of the center of the top of the left bridge tower towards the mid-span after deformation is expressed as:

[0241] △ B = (H 1,u + H 1,d - H L,u - H L,d )·δ B

[0242] In the formula, δ B is the lateral bending flexibility coefficient of the left tower, m / kN; b t is the horizontal distance between each tower column and the central axis H1,u and H 1,d are the horizontal component forces of the first - stage main cables at the upstream and downstream of the main span respectively; H L,u and H L,d are the horizontal component forces of the main cables at the upstream and downstream of the left side - span respectively.

[0243] The torsional angle (positive in the counter - clockwise direction) θ of the left pylon after deformation B is expressed as:

[0244] θ B =(H 1,d - H L,d - H 1,u +H L,u )b t ·ζ B

[0245] In the formula, ζ B is the torsional flexibility coefficient of the left pylon; b t is the horizontal distance between each tower column and the central axis.

[0246] The longitudinal offsets at the tops of the upstream and downstream tower columns of the left pylon are respectively:

[0247] △ B,u =△ B - θ B ·b t

[0248] △ B,d =△ B +θ B ·b t

[0249] The offset Δ of the center of the top of the right pylon towards the mid - span after deformation C is expressed as:

[0250] △ C =(H n,u +H n,d - H R,u - H R,d )·δ C

[0251] In the formula, δ C is the lateral bending flexibility coefficient of the right pylon, m / kN; H n,u and H n,d are the horizontal component forces of the n - th stage main cables at the upstream and downstream of the main span respectively; H R,u and H R,d are the horizontal component forces of the main cables at the upstream and downstream of the right side - span respectively.

[0252] The torsional angle (positive in the counter - clockwise direction) θ of the right pylon after deformation C is expressed as:

[0253] θ C =(H R,d -H n,d -H R,u +H n,u )b t ·ζ C

[0254] In the formula, ζ C is the torsional flexibility coefficient of the right tower.

[0255] The longitudinal offsets of the top of the upstream and downstream tower columns of the right tower are respectively:

[0256] △ C,u =-△ C -θ C ·b t

[0257] △ C,d =-△ C +θ C ·b t

[0258] Step 6: Express the magnitudes of the three directions of the suspender and the spatial position coordinates of the lower suspension point in terms of the basic unknowns, including the longitudinal, transverse, and vertical coordinates.

[0259] Under the combined action of the dead load and the live load, the coordinates of the lower suspension point of any suspender in the global coordinate system can be expressed as:

[0260] X b,i =x i +v

[0261] Y b,i =±b h -△y b,i

[0262]

[0263] In the formula, X b,i , Y b,i and Z b,i are the X, Y, and Z coordinates of the i-th lower suspension point in the global coordinate system; v is the longitudinal rigid body displacement of the main girder; Δy b,i is the transverse deflection of the main girder at the i-th pair of suspenders; Δz b,i is the vertical deflection of the main girder at the i-th pair of suspenders; the ± here are applicable to the upstream and downstream respectively; H B is the elevation of the left bridge tower.

[0264] The transverse inclination angle θ i and the vertical inclination angle As Figure 11 shown, can be expressed as follows respectively:

[0265]

[0266]

[0267] The component forces of the suspension rod force in three directions can be respectively expressed as:

[0268]

[0269]

[0270]

[0271] Therefore, the spatial position of the suspension rod can be expressed by the basic unknowns.

[0272] Step 7: Establish control equations equal to the number of basic unknowns, including the conservation of the stress-free length of each catenary segment of the main cable; the conservation of the stress-free length of each suspension rod; the closure of the span and elevation difference of each span; and the force balance of the main girder.

[0273] (1) Conservation of the stress-free length of each catenary segment of the main cable:

[0274] S c,L = S' c,L

[0275] S c,i = S' c,i , 1 ≤ i ≤ n + 1

[0276] S c,R = S' c,R

[0277] In the formula, S' c,L and S' c,R are the stress-free lengths of the main cables of the left and right side spans in the initial state respectively; S' c,i is the stress-free length of the i-th catenary segment of the main cable in the main span in the initial state; S h,i is the stress-free length of the i-th suspension rod after the combined action of the dead load and the live load.

[0278] (2) Conservation of the stress-free length of each suspension rod:

[0279]

[0280] S h,i = S' h,i

[0281] In the formula, E h and A h are the elastic modulus and cross-sectional area of the suspension rod respectively; S' h,i is the stress-free length of the i-th suspension rod in the initial state.

[0282] (3) Taking the upstream side as an example, the span lengths and elevation differences of each span are closed. By changing u to d in the following formula, the control equations for the downstream side can be obtained:

[0283] Closure of the span lengths of each span:

[0284] L′ 1,u = L 1,u -△ B,u

[0285]

[0286] L′ 3,u = L 3,u -△ C,u

[0287] In the formula, L′ 1,u , L′ 2,u and L′ 3,u are the horizontal projection lengths of the catenary of the left side span, main span, and right side span respectively after being subjected to the lateral load.

[0288] Closure of the elevation differences of each span (taking the upstream side as an example):

[0289] △z L,u = H B - H A

[0290]

[0291] △z R,u = H C - H D

[0292] Closure of the horizontal distance in the transverse direction between the two endpoints of the main cable of the main span (taking the upstream side as an example):

[0293]

[0294] (4) Force balance of the main girder:

[0295] After the main girder is subjected to the combined action of the dead load and live load, the force balance in the x, y, and z directions:

[0296]

[0297]

[0298]

[0299] After the main girder is subjected to the combined action of the dead load and live load, the moment balance about the x, y, and z axes:

[0300]

[0301]

[0302]

[0303] The torsional angle of the right end of the main girder about the x-axis is 0:

[0304]

[0305] Step 8: Solve the control equations to obtain the values of all basic unknowns.

[0306] A total of 4n + 27 control equations were obtained in step (7). Move the right ends of each equation to the left end of the equation to obtain a form like f i = 0 (i = 1, 2... n), and then rewrite it as a sum of squares to obtain a form like f i 2 = 0 (i = 1, 2... n), and finally add them up to obtain the objective function:

[0307]

[0308] Use the method of programming solution to perform programming solution on the objective function, and solve the values of the 4n + 27 basic unknowns in step (7) so that the 4n + 27 control equations in step (2) hold simultaneously.

[0309] Step 9: Substitute the basic unknowns into the above steps (2), (3), (4), (5), and (6) to obtain the full-bridge response.

[0310] Substitute the values of the basic unknowns obtained in step (8) back into step (3) to determine the response of the main cable under live load, including the coordinates of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge;

[0311] Substitute the values of the basic unknowns obtained in step (8) back into step (4) to determine the response of the main girder under live load, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder;

[0312] Substitute the values of the basic unknowns obtained in step (8) back into step (5) to determine the deformation and internal force of the suspenders under live load;

[0313] Substitute the values of the basic unknowns obtained in step (8) back into step (6) to determine the position and spatial position of the suspenders under live load, including the offsets of the suspenders in the longitudinal, transverse, and vertical directions.

Claims

1. A method for determining the structural deformation and internal forces of a suspension bridge under the action of laterally distributed live loads, characterized in that: It includes the following steps: Step 1: Identify all the internal force and deformation characteristic parameters of the suspension bridge in the dead load state; all the internal force and deformation characteristic parameters in Step 1 mainly include the following three aspects: (1) Establish geometric form coordinates; including the upper and lower suspension point coordinates of the suspenders, the top coordinates of the bridge towers, the cable anchor point coordinates, the side span and main span lengths, the number and spacing of the suspenders, and the cable sag-span ratios of each span; (2) Set internal force parameters; including the internal forces of the main cable, main girder and bridge tower, and the suspender forces; (3) Set physical characteristics; including the lateral bending and torsional flexibility coefficients of the bridge tower, the self-weight per unit length, elastic modulus, moment of inertia, cross-sectional area, Poisson's ratio, and density of each component; Step 2: Determine the independent basic unknowns representing the key geometric forms and internal force parameters of the entire bridge; The basic unknowns in Step 2 include: the catenary equation parameters a L,u and a L,d for the left side span of the main cable, the catenary equation parameters a 1,u and a 1,d for the first catenary segment of the main span of the main cable, and the catenary equation parameters a R,u and a R,d for the right side span of the main cable; the horizontal projection lengths L 1,u and L 1,d of the catenary segments for the left side span of the main cable, the horizontal projection lengths l 1,u to l n+1,u and l 1,d to l n+1,d of the catenary segments for the first catenary segment of the main span of the main cable, and the horizontal projection lengths L 3,u and L 3,d of the catenary segments for the right side span of the main cable; the horizontal component forces H L,u and H L,d of the main cable for the left side span, the horizontal component forces H 1,u and H 1,d of the main cable for the first catenary segment of the main span, and the horizontal component forces H R,u and H R,d of the main cable for the right side span; the lateral deflection angles α 1,u and α 1,d of the main cable for the first catenary segment of the main span; the hanger forces P 1,u to P n,u and P 1,d to P n,d ; the change in the vertical support reaction of the main beam ΔF L,z,u and ΔF L,z,d and ΔF R,z,u and ΔF R,z,d ; the horizontal support reaction F L,y and F R,y of the wind-resistant support of the main beam; the horizontal rigid body displacement ν of the main beam in the longitudinal direction of the bridge; where the subscript "u" represents upstream, "d" represents downstream, and "n" represents the number of hangers, with a total of 4n + 27 independent basic unknowns; Step 3: Express the main cable response using the basic unknowns, including the coordinates of the suspension points on the main cable in the longitudinal, vertical, and transverse directions of the bridge; Step 4: Express the response of the main girder using the basic unknowns, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder; Step 5: Express the lateral displacement and torsional deformation of the bridge tower using the basic unknowns; Step 6: Express the suspender response using the basic unknowns, including the magnitudes of the forces in the three directions of the suspenders and their longitudinal, transverse, and vertical coordinates; Step 7: Establish control equations equal in number to the number of basic unknowns, including the conservation of the stress-free length of each catenary segment of the main cable; the conservation of the stress-free length of each suspender; the closure of the span and elevation difference of each span; and the force balance of the main girder; Step 8: Solve the control equation system to obtain the values of all the basic unknowns; Step 9: Substitute the basic unknowns into Steps 2, 3, 4, 5, and 6 above, and then obtain the response of the entire bridge.

2. The method for determining the structural deformation and internal forces of a suspension bridge under the action of laterally distributed live loads according to claim 1, characterized in that, In Step 3, expressing the main cable response using the basic unknowns, including the coordinates of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge, and establishing an overall coordinate system with the torsional center of the cross-section at the left end of the main girder in the initial state as the origin, the specific steps are as follows: (3.1) The elevation difference between the head and tail endpoints of the catenary of the left side span main cable is: where c L = -H L / q, H L is the horizontal component force of the main cable in the left span under the combined action of dead load and live load; q is the self-weight per unit length of the main cable, kN / m; a L is the parameter of the catenary equation in the left span; L 1 is the horizontal projection length of the main cable in the left span. The stress-free length of the left side span main cable is: where: E c and A c are the elastic modulus and cross-sectional area of the main cable, respectively; (3.2) The elevation difference between the head and tail endpoints of the catenary of the right side span main cable is: where c R = -H R / q, H R is the horizontal component force of the main cable in the right span under the combined action of dead load and live load; a R is the parameter of the catenary equation for the right span; L 3 is the horizontal projection length of the main cable in the right span; The stress-free length of the right side span main cable is: (3.3) The elevation difference between the head and tail endpoints of the i-th catenary segment of the main span main cable is: where α i is the angle between the plane where the i-th catenary is located and the xoz plane; c i = -H i / q, where H i is the horizontal component force of the i-th main cable under the combined action of dead load and live load; a i is the parameter of the i-th catenary equation of the main span; l i is the horizontal projection length of the i-th main cable of the main span. The stress-free length of the main cable of the i-th catenary segment can be expressed as: (3.4) Under the combined action of dead load and live load, the recurrence relationships of the catenary parameter a, the horizontal cable force H, and the angle α between the plane where the catenary is located and the xoz plane between adjacent catenary segments of the main span are expressed as: Wherein, P i,x , P i,y , P i,z are respectively the component forces of the suspension rod force P i in the x, y, and z directions; Therefore, given the a, H, and α of the first catenary segment of the main span, the catenary equation parameters of the remaining segments can all be expressed; the catenary parameters of each segment of the main cable can all be represented by the basic unknowns; (3.5) The coordinates of the suspension points on the main cable: Y c,i = ±b h -Δy c,i Wherein, X c,i , Y c,i and Z c,i are respectively the X, Y, and Z coordinates of the i-th upper suspension point in the global coordinate system; Δx B is the offset of the bridge tower; b h is the transverse distance from the suspender to the longitudinal center line of the main girder; H B is the elevation of the left bridge tower.

3. The method for determining the structural deformation and internal forces of a suspension bridge under the action of laterally distributed live loads according to claim 1, characterized in that, In Step 3, the responses of the main girder are expressed in terms of the basic unknowns, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder. The specific steps are as follows: (4.1) Vertical deflection of the main girder: The main beam is divided into n + 1 segments along the bridge axis by the suspender cables. Vertically, the main beam is subjected to a total of n + 2 forces, including the incremental vertical reaction forces ΔF L,z 、ΔF R,z and n pairs of incremental vertical suspender forces. At the m-th suspender, the vertical deflection of the main beam can be expressed as: where x m represents the horizontal distance of the m-th pair of suspenders from the left end of the main girder; x i represents the horizontal distance of the i-th pair of suspenders from the left end of the main girder; ΔP i,z represents the increment of the vertical suspender force of the i-th pair of suspenders; E b is the elastic modulus of the main girder; I z is the vertical flexural moment of inertia of the main girder; C m,1 and C m,2 represent the constant terms after integrating the vertical bending moment of the main girder; The indefinite integral constant C of the m-th main girder and the (m + 1)-th main girder m,1 and C m+1,1 as well as C m,2 and C m+1,2 The recurrence relation is as follows: where, ΔP m,z represents the vertical hanger force increment of the m-th pair of hangers; The indefinite integral constants of the first segment of the main girder are as follows: C 1,2 ＝0 (4.2) Transverse deflection of the main girder: Along the longitudinal direction of the bridge, the main girder is divided into n + 1 segments by suspenders, and the main girder receives a total of n + 2 forces in the transverse direction of the bridge; including the transverse reaction forces F L,y , F R,y and n pairs of transverse suspender force increments; at the m-th suspender, the transverse deflection of the main girder can be expressed as: where ΔP m,y represents the increment of the lateral hanger force of the m-th pair of hangers; I y is the lateral flexural moment of inertia of the main girder; D m,1 and D m,2 represent the constant terms after the integration of the lateral bending moment of the main girder; κ(x) is a function generated by the double integration of the bending moment; The indefinite integral constant D of the m-th main beam and the (m + 1)-th main beam m,1 and D m+1,1 as well as D m,2 and D m+1,2 The recurrence relation is as follows: where ΔP m,y represents the increment of the lateral hanger force of the m-th pair of hangers; The indefinite integral constants of the first segment of the main girder are as follows: D 1,2 ＝0 (4.3) Torsional deformation of the main girder: Due to the transverse deflection of the main girder, the vertical components of the hanger forces upstream and downstream are no longer symmetric. The main girder will twist under the action of asymmetric vertical hanger forces. The torque within the m-th segment of the main girder is: where b s and b h are the transverse distances from the vertical supports at the beam ends and each suspender to the longitudinal center line of the main beam respectively; ΔP i,z,u and ΔP i,z,d are the suspender forces at the upstream and downstream of the i-th suspender of the main beam in the longitudinal direction of the bridge respectively; The rotation angle of the right end of the m-th segment of the main girder relative to the left end of this segment is: where d m is the length of the m-th main girder; G is the shear modulus of the main girder; I P is the polar moment of inertia of the main girder; The vertical deflections caused by torsion at the i-th suspension point on the upstream and downstream sides of the main beam are respectively and The transverse and vertical deflections of the main girder can both be expressed in terms of the basic unknowns.

4. A method for determining the structural deformation and internal forces of a suspension bridge under the action of a laterally distributed live load according to claim 1, characterized in that: In Step 5, the lateral displacement and torsional deformation of the bridge tower are expressed in terms of the basic unknowns. The specific steps are as follows: The offset Δ of the center of the left pylon top towards the mid-span after deformation B It is expressed as: Δ B =(H 1,u +H 1,d -H L,u -H L,d )·δ B where δ B is the lateral bending flexibility coefficient of the left tower, m / kN; H 1,u and H 1,d are respectively the horizontal component forces of the first main cable section upstream and downstream of the main span; H L,u and H L,d are respectively the horizontal component forces of the main cable upstream and downstream of the left side span. The torsional angle θ of the left pylon after deformation B It is expressed as: θ B = (H 1,d - H L,d - H 1,u + H L,u )b t ·ζ B where ζ B is the torsional flexibility coefficient of the left tower; b t is the horizontal distance between each tower column and the central axis; The longitudinal offsets of the tops of the upstream and downstream tower columns of the left tower are respectively: Δ B,u = Δ B - θ B · b t Δ B,d = Δ B + θ B · b t The offset Δ of the center of the right pylon top towards the mid-span after deformation C It is expressed as: Δ C =(H n,u +H n,d -H R,u -H R,d )·δ C where δ C is the lateral bending flexibility coefficient of the right tower, m / kN; H n,u and H n,d are the horizontal component forces of the main cable in the nth section upstream and downstream of the main span respectively; H R,u and H R,d are the horizontal component forces of the main cable upstream and downstream of the right side span respectively. The torsional angle θ of the right pylon after deformation C It is expressed as: θ C = (H R,d - H n,d - H R,u + H n,u )b t ·ζ C where ζ C is the torsional flexibility coefficient of the right tower; The longitudinal offsets of the tops of the upstream and downstream tower columns of the right tower are respectively: Δ C,u = -Δ C -θ C ·b t Δ C,d = -Δ C + θ C · b t The responses of the bridge tower can all be expressed in terms of the basic unknowns.

5. A method for determining the structural deformation and internal forces of a suspension bridge under the action of a laterally distributed live load according to claim 1, characterized in that: In Step 6, the magnitudes of the three-direction components of the hanger force and the longitudinal, transverse, and vertical coordinates of the lower suspension points are expressed in terms of the basic unknowns. The specific steps are as follows: Under the combined action of the dead load and the live load, the coordinates of the lower suspension point of any hanger in the global coordinate system can be expressed as: X b,i = x i + v Y b,i = ±b h -Δy b,i where X b,i , Y b,i and Z b,i are the X, Y, and Z coordinates of the i-th lower suspension point in the global coordinate system, respectively; v is the longitudinal rigid body displacement of the main girder; Δy b,i is the lateral deflection of the main girder at the i-th pair of suspenders; Δz b,i is the vertical deflection of the main girder at the i-th pair of suspenders; the ± here applies to the upstream and downstream, respectively; H B is the elevation of the left bridge tower; The lateral inclination angle θ of the suspension rod i and the vertical inclination angle can be respectively expressed as follows: The components of the hanger force in the three directions can be respectively expressed as: Therefore, the magnitudes of the three-direction components of the hanger force and the spatial coordinates can be expressed in terms of the basic unknowns.

6. A method for determining the structural deformation and internal forces of a suspension bridge under the action of a laterally distributed live load according to claim 1, characterized in that, In Step 7, control equations equal in number to the basic unknowns are established, including the conservation of the stress-free length of each catenary segment of the main cable; the conservation of the stress-free length of each hanger; the closure of the span and elevation difference of each span; and the force balance of the main girder. The specific steps are as follows: (7.1) Conservation of the stress-free length of each catenary segment of the main cable: S c,L = S c ′ ,L S c,i = S c ′ ,i , 1 ≤ i ≤ n + 1 S c,R = S c ′ ,R Wherein, S c ′ ,L and S c ′ ,R are respectively the stress-free lengths of the main cables of the left and right side spans in the initial state; S c ′ ,i is the stress-free length of the i-th catenary section of the main cable of the main span in the initial state; (7.2) Conservation of the stress-free length of each hanger: S h,i = S h ′ ,i where, E h and A h are respectively the elastic modulus and cross-sectional area of the suspension rod; S h ′ ,i is the stress-free length of the i-th suspension rod in the initial state; S h,i is the stress-free length after the combined action of the dead load and the live load; (7.3) For the upstream side, the closure of the span and elevation difference of each span: Closure of the span of each span: L 1 ′ ,u = L 1,u - Δ B,u L 3 ′ ,u = L 3,u - Δ C,u where L 1 ′ ,u , L 2 ′ ,u and L 3 ′ ,u are respectively the catenary horizontal projection lengths of the left span, main span and right span after being subjected to lateral loads; For the upstream side, the closure of the elevation difference of each span: Δz L,u = H B - H A Δz R,u = H C - H D Closure of the transverse horizontal distance between the two endpoints of the main cable of the main span. For the upstream side: (7.4) Force balance of the main girder: After the main girder is under the combined action of the dead load and the live load, the force balance in the x, y, and z directions: After the main girder is under the combined action of the dead load and the live load, the moment balance about the x, y, and z axes: The torsional angle of the right end of the main girder about the x axis is 0: So far, a total of 4n + 27 independent control equations have been listed.

7. A method for determining the structural deformation and internal forces of a suspension bridge under the action of a laterally distributed live load according to claim 1, characterized in that, In Step 8, the control equation system is solved to obtain the values of all the basic unknowns. The specific steps are as follows: A total of 4n + 27 control equations are obtained in Step 7; move the right - hand side of each equation to the left - hand side of the equation to get the form of f i = 0, where i = 1, 2... n, and then rewrite it as the sum of squares to get f i 2 = 0, where i = 1, 2... n, and finally add them up to obtain the objective function: Using the method of solving optimization problems, the objective function is optimized to solve for the values of the 4n + 27 basic unknowns in Step 7.

8. A method for determining the structural deformation and internal forces of a suspension bridge under the action of a laterally distributed live load according to claim 1, characterized in that, Step 9: Substitute the basic unknowns into Steps 2, 3, 4, 5, and 6 above, and then obtain the responses of the entire bridge. The specific steps are as follows: Substitute the values of the basic unknowns obtained in Step 8 back into Step 3 to determine the responses of the main cable under the action of the live load, including the displacements of the suspension points of the main cable in the longitudinal, vertical, and transverse directions of the bridge; Substitute the values of the basic unknowns obtained in Step 8 back into Step 4 to determine the responses of the main girder under the action of the live load, including the longitudinal drift, transverse deflection, vertical deflection, and torsional deformation of the main girder; Substitute the values of the basic unknowns obtained in Step 8 back into Step 5 to determine the deformations and internal forces of the suspenders under the action of the live load; Substitute the values of the basic unknowns obtained in Step 8 back into Step 6 to determine the spatial positions of the suspenders under the action of the live load, including the offsets of the suspenders in the longitudinal, transverse, and vertical directions.

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